Briançon–Skoda theorem

Remarks

Joël Briançon and Henri Skoda proved the theorem in 1974, using Skoda's $L^2$ estimates for the division problem. A modern proof uses multiplier ideals: one has $\overline{I^{\,l}} \subseteq \mathcal{J}(I^{\,l})$ for all $l$, and Skoda's theorem gives $\mathcal{J}(I^{\,l}) \subseteq I \cdot \mathcal{J}(I^{\,l-1})$ for $l \ge n$, so induction gives the inclusion. Lipman and Teissier, and Lipman and Sathaye, extended the result to regular local rings.

A corollary answers a question of Mather. If $f \in \mathcal{O}_n$ and $f(0) = 0$, then $f$ lies in the integral closure of its Jacobian ideal $J(f) = (\partial f / \partial z_1, \dots, \partial f / \partial z_n)$, and hence $f^n \in J(f)$.

The exponent cannot be lowered to $1$ in general. For $n = 2$ and $I = (x^2, y^2)$, the element $xy$ satisfies $(xy)^2 - x^2 y^2 = 0$ with $x^2 y^2 \in I^2$, so $xy \in \overline{I}$, but $xy \notin I$. The theorem gives instead $\overline{I^2} \subseteq I$.

Going deeper

For a student

In analytic terms, $h \in \overline{I}$ when $\abs{h} \le C (\abs{g_1} + \cdots + \abs{g_r})$ near $0$, where $g_1, \dots, g_r$ generate $I$. The theorem turns such a size bound into an algebraic membership in a power of $I$. This is how it is used in the study of the Jacobian ideal and of the Milnor number. It is a model for many effective results in commutative algebra and for applications of $L^2$ methods in algebraic geometry.